Sparsity of postcritically finite maps of ℙ k and beyond: A complex analytic approach

An endomorphism f : ℙ k → ℙ k of degree d ≥ 2 is said to be postcritically finite (or PCF) if its critical set Crit ( f ) is preperiodic, i.e. if there are integers m > n ≥ 0 such that f m ( Crit ( f ) ) ⊆ f n ( Crit ( f ) ) . When k ≥ 2 , it was conjectured in [61] that, in the space End d k of all endomorphisms of degree d of ℙ k , such endomorphisms are not Zariski dense. We prove this conjecture. Further, in the space Poly d 2 of all regular polynomial endomorphisms of degree d ≥ 2 of the affine plane 𝔸 2 , we construct a dense and Zariski open subset where we have a uniform bound on the number of preperiodic points lying in the critical set. The key object in the article are the complex bifurcation measure and its properties. The proofs are a combination of the theory of heights in arithmetic dynamics and methods from real dynamics to produce open subsets with maximal bifurcation.